In this explainer, we will learn how to identify, write, and evaluate a logarithmic function as an inverse of the exponential function.
A logarithmic function is just the inverse of an exponential function. Before looking at logarithmic functions, however, let’s consider a linear function, such as , and its inverse. Recall that to find the function’s inverse, we would first rewrite it as . Then, we would exchange the variables and to get and solve for , giving us . Our calculations reveal that the inverse of is . You can also define the inverse as . Since and are inverses of each other, if the point satisfies , then the point must satisfy . For example, we can see that the point satisfies because and that the point satisfies , because
Examining the graphs of and for the two functions below, we can see that they are reflections of each other in the line .
Now let’s consider the exponential function . The inverse of this function is the logarithmic function or . Suppose we are asked to find for the exponential function . We would substitute 1 for to get . Next, suppose we are asked to find for the logarithmic function . We would substitute 5 for to get and ask ourselves the following question: “What power is a base of 5 raised to in order to equal 5?” Since the answer to the question is 1, we know that . Notice that the point satisfies the exponential function, while the point satisfies the logarithmic function. Just as with the linear function and its inverse above, the coordinates of the points that satisfy the two functions are reversed, and the graphs of the two functions are reflections of each other in the line as shown.
This will be true for any base, , of an exponential function and its inverse logarithmic function.
Definition: Logarithmic Function
A logarithmic function is the inverse of an exponential function. For the exponential function , its inverse logarithmic function is or . If the point satisfies the exponential function, then the point satisfies the logarithmic function. That is, if , then . The graphs of the two functions are reflections in the line .
Keep in mind that according to this definition, the exponential function would have an inverse logarithmic function of or . When the base is 10, however, by convention, there is no need to specify it in the logarithmic function. That is, we can just write so that is taken to be (which can be read as log base 10 of or, simply, as log of ). Likewise, for the exponential function , the inverse logarithmic function needs to be written in a special way. Instead of writing or , we would write or (which can be read as the natural log of ).
Definition: Natural Logarithmic Function
A natural logarithmic function is the inverse of an exponential function with a base of . Given that , its inverse natural logarithmic function is or .
Now let’s look at some problems related to logarithmic functions.
Example 1: Finding the Inverse Logarithmic Function
The function has an inverse in the form . What are the values of and ?
Answer
Recall that when finding the inverse of a linear function, we exchange the variables and and then solve for . To find the inverse logarithmic function, we need to follow the same procedure. Let’s begin by rewriting the given exponential function as . After exchanging the variables and , we get . Subtracting 3 from both sides of the equation gives us , and then dividing both sides by 2 gives
Now, since a natural logarithmic function has a base of , let’s take the natural log of both sides of the equation. After rewriting the equation as we can ask ourselves the following question to simplify the right side: “What power is a base of raised to in order to equal ?” The answer to the question is , so the equation can be rewritten as or . We can then replace with to get and rewrite the function as to put it in the form . This shows that and .
Note
Remember that if the point satisfies an exponential function, then the point satisfies its inverse logarithmic function. Let’s find a point that satisfies and check to see if the point satisfies . If satisfies , it will not prove that our answer is correct, but if does not satisfy the function, we will know for certain that we made a mistake.
Since , the point satisfies . This means that the point should satisfy . We can determine if this is the case by finding as follows:
This shows that the point does, in fact, satisfy , as it should.
In the next example, we will demonstrate the relationship between the domain and range of an exponential function and the domain and range of its inverse. Recall that if the point satisfies an exponential function, then the point satisfies its inverse logarithmic function. Thus, if is an element of the domain of the exponential function, it is also an element of the range of the logarithmic function. Likewise, if is an element of the range of the exponential function, then it is also an element of the domain of the logarithmic function. This is true for any point , so we know that the domain of the exponential function must be the same as the range of the logarithmic function. Likewise, the range of the exponential function must be the same as the domain of the logarithmic function.
Example 2: Finding the Domain of the Inverse of an Exponential Function
Consider the function , where is a positive real number not equal to 1. What is the domain of ?
Answer
Recall that the domain of a function is the set of all possible input values, and the range of a function is the set of all possible output values. First, let’s consider the domain and range of the function . Since the exponent in the function’s definition can be any negative value, any positive value, or 0, the function’s domain is all real numbers. We were given that is positive, so to help determine the function’s range, let’s take a specific positive value to use as an example, , which would give us the function . A negative value of , such as , gives ; a positive value of , such as 3, gives ; and a value of 0 for gives . Notice that the output value in each case is positive, so we know that the range of must be .
Since the exponent in is a variable, we also know that the function is an exponential function. Recall that the inverse of an exponential function is a logarithmic function. That is, if , then . Also recall that the range of an exponential function is the domain of its inverse. In other words, the domain of the logarithmic function must be .
Note
We can verify our answer by again assuming that and by then graphing both and for the functions and as follows:
We can see that the graphs are reflections of each other in the line and that the graph of is located in only the first and fourth quadrants. In other words, it has the -axis as an asymptote and has only positive input values. This confirms that the domain of the function is, in fact, .
Now let’s look at how we can evaluate a logarithmic function.
Example 3: Evaluating a Logarithmic Function at a Given Point
Consider the function . If , find the value of .
Answer
To find the value of , we can begin by substituting into the given function for and 3 for to get
Recall that a logarithmic function is the inverse of an exponential function and that if , then . It follows that if , then .
We can see that in this problem, the base is 2, the value of is 3, and the value of is . Substituting these values into the equation gives us .
Simplifying gives , and solving for gives a solution of .
Note
By finding for the function , we can check our answer. Substituting 3 for gives us . After multiplying 3 by 3, we get , and after subtracting 1 from 9, we get . To simplify the right side of this equation, we must ask ourselves the following question: “What power is a base of 2 raised to in order to equal 8?“ The answer is 3, so we know that and that our answer is correct.
In the next example, we will determine the base of a logarithmic function given a point that the function’s graph passes through.
Example 4: Completing a Function Using a Given Point
Given that the graph of the function passes through the point , find the value of .
Answer
To find the value of , first we need to rewrite the logarithmic function as . Since the graph of the function passes through the point , we know that when , then . This allows us to substitute these values into the function to get the equation
We know that if , then , so it follows that . One way to solve this equation for is to take the fifth root of each side as follows:
This shows that the value of is 4. However, without a calculator, it may be difficult for us to determine that the fifth root of 1 024 is 4. One strategy we might employ to find the fifth root of 1 024 is to recognize that 1 024 is a power of 2. We can list the powers of 2 as follows:
With this information, we can solve the equation for by substituting for 1 024 and grouping the 2’s as shown:
This method gives us the same value of 4 for that we got previously.
As a final example, let’s look at a real-world problem.
Example 5: Solving a Real-World Problem Using Logarithmic Functions
The pH of a solution is given by the formula , where is the concentration of hydrogen ions. Determine the concentration of hydrogen ions in a solution whose pH is 8.4.
Answer
The concentration of hydrogen ions is represented by , so we must solve for this variable to answer the question. Since we are given that the pH of the solution is 8.4, we can begin by substituting 8.4 into the formula for the pH to get
After substituting, we can then multiply both sides of the equation by to get . Recall that when the base of a logarithm is not shown, it is assumed to be 10, so in order to help solve for , we can now rewrite the equation as
We know that a logarithmic function is the inverse of an exponential function and that if , then , so based on the information we have, we can write an exponential equation by substituting values or variables into for , , and . Since , , and , we get the equation
This shows that the concentration of hydrogen ions in a solution whose pH is 8.4 is .
Recall that the domain of a logarithmic function is , so in this case, must be a positive number. Its value is, in fact, positive because 10 raised to any power is greater than or equal to 0. The negative exponent in only means that the value of is less than 1. By using a calculator, we can see that its approximate value is or 0.00000000398.
Now let’s finish by recapping some key points.
Key Points
- A logarithmic function is the inverse of an exponential function.
- For the exponential function , its inverse logarithmic function is or .
- When the base of a logarithmic function is 10, there is no need to specify it. If , then .
- A natural logarithmic function is the inverse of an exponential function with a base of . If , then .
- The graphs of an exponential function and its inverse logarithmic function are reflections in the line .
- The domain of a logarithmic function is , which is the range of its inverse exponential function.
